More: en.fufaev.org/advanced-maths Books by Alexander Fufaev: 1) Equations of Physics: Solve EVERY Physics Problem en.fufaev.org/physics-equations-book 2) Alexander Fufaev and His Never Ending Story: en.fufaev.org/lifestory
@@savejobarIn US it's generally called Calculus 3 or Multivariable calculus or vector calculus. The generalized Stokes' Theorem is usually covered in a course called Differential Geometry
The very essence of clarity. As a city bus driver, I must say this is easily the most straightforward explanation of GDT I have ever heard. Many thanks!
As I read this, all of the connotations of these words arranged this way is almost absurd in what it says: It starts with the hyperbolic “very essence of clarity.” What do you or anyone else generally assume about someone who is a bus driver? If it isn’t that why is it relevant enough to mention? In addition to above, would we assume that any random or even average bus driver would have heard of, knows or thought about GDT?
@@compegord07 I was just having some fun with some of the other commenters here. Several have said things like, “I’m an Electrical Engineer and this is best explanation …” and “I am a PhD student in Mathematics, and this clearest video …” I really am a bus driver, and although I do enjoy watching videos about math, this was just above my level. I certainly don’t pretend to speak for all bus drivers.
It is so amazing!!!!!!!!!!!. Unlike the book just teaching us how to put the number into it, it explains what exactly we are doing so we can be more easy to understand and accept the concept. Thank you so much.
I was at a taster lecture where a Cambridge professor tried to explain this formula, and I was entirely dumbfounded. But now that I've watched your video, it all makes sense!
You'll get scalars, vectors, and partial derivatives in Linear Algebra. You'll ace the first exam. Then, the rest of the semester or quarter you'll be cussing out your computer. 🥴
Have Faith in yourself! If you do not understand then ASK! Any Prof who looks down at your question is a failed Prof. We all have to learn, it's just that some prof's get so comfortable that they forget their own troublesome journey to tenure. I came from the wrong side of the tracks and did mathematics and upset a few with my up front, no respect, questioning. Luckily there were enough Profs around to shield me from expulsion for being rude in response to a ridiculous answer. What I am saying is that You have the ability to achieve, don't let that be curtailed by a numpty with a superiority complex.
Physicists: “The sum of the forces entering or leaving an imaginary spacial boundary is equivalent in magnitude and direction to the sum of the forces at the boundary itself” College students: “This is getting complicated.”
the idea is that it sums up to only what’s on the boundary surface of a volume, because all internal small volume parts dv mutually compensate (as to 0) each other on adjacent neighboring surfaces
This is really how one should explain the divergence theorem . It plays a central role in partial differential equations and has many applications. Well done !
Wish I could like this video twice. I know nothing of the intuition behind Electrodynamics, and yet this made it max’s equations seem as clear as day. Great work!
Thank you for your explanation! It's the clearest explanation of the divergence theorem I've ever seen. When reading the calculus textbook, it just tell me the theorem and a mathematical proof, but the connection between the divergence theorem and the green theorem is still ambiguous to me. Your animation helps me understand the connection between them! Thank you very much!
This is a gem - I came across it as a suggestion from RU-vid while viewing tutorials about Transformers. The equation in this video gave me a deeper understading of the use of scalar and dot product in understanding the similarties between features and the relationships between features. Thanks a lot!
Amazing video! So much explained and illustrated beautifully and detailedlly. A real deep explain for each component in the writing convention. Keep on the exellent work!
I am an EE engineer and I must say, this is THE BEST illustration of GDT I have seen, and I have seen MANY! Well done. (Same with your Maxwell Equation video)
the thumbnail does a really good job of relating the divergence integral with the closed loop one i imagine the cubes within the larger cube to represent the infinitesimal points at which you test divergence (it is based off of a derivative after all), and if you just project the surface faces of the infinitesimals to the larger box you get the flux thru the entire surface
@leinyuymarie5570 I'll take that a step further (attending a class where I'm lost). When I was in my junior year working on my BSEE, EMAG was (and I'm sure still is) required. Like most engineering classes, they throw math at you and wave their hands like a Jedi to make it all go away and you're supposed to absorb the subject matter conceptually. At UMASS (Amherst), they broke the course into two semesters, but places like MIT and other heavyweight universities do it in one. I never failed a course in my life, but I failed the second semester. Never had I put so much effort into a class and failed, I was nearly suicidal. I made my way to my car parked in the student lot and shamelessly sobbed uncontrollably for at least an hour. I had just finished 6 years of abuse in the Navy prior to starting this BSEE quest, and they couldn't make me cry, but here I was crying like a baby. Vector Calculus is no joke, so there's no shame in not "getting it". In fact, I understood the phenomena very well having covered the the subject matter (electric and magnetic flux) in lower division physics where you can see it all happen in front of you and also in Calculus-III, yet another course where MIT covers it all in a single semester, others in 2, but I had to take it in a 2-yr SUNY school where they broke it in 3 because 6-yrs was just too long a lay-off to just jump into calculus, so I dropped Calc-II at UMASS and they allowed me to trade Calc-II and III from an "acceptable" program at another university. It was a classic, "fake-it till you make-it. This reply is now way longer than I intended, so I'll wrap it up. Since I failed EMAG-II, it meant that my 4-yr degree would now take 5-yrs. A financial ouchie and further embarrassment of being 30 yrs old before completing the program. I mentally shelved the subject matter for year (not needing it for anything) and (after a little prep work) took a vector calculus course in the math department where they did a far better job covering the material just like Alexander does here in this YT Vid. In my last semester (10 of 10) I took EMAG-II and got an A. It helped that there was a much better professor for that go-round, and the Vector Calc class sure helped. It just took some time. So cheer up, you earned your seat to get into that class, so you are capable understanding the material. There was no internet when I failed, so you have the advantage of access to great vids like this to help you to wrap your head around the material. I'm trying to help my nephew through it right now and sure would have a hard time without great vids like this. You Can Do It!!!
so my guess is that from the perspective of the volume integral, you're splitting the enclosed volume into like infitesimally small cubes, and when you're adding them up, every two adjacent cubes share one face, and both bring it up into the computations but with opposite effects (facing opposite directions), so the inner stuff all gets cancelled out and you end up only with the outer faces, which is just the surface?
You don't even need to got that far. The volume integral sums up the sources and sinks that "create" positive and negative flux. Inside that volume. The remaining flux has to flow out of (or into respectively) the volume. Picture each source as a tap and each sink as a drain. If all taps combined carry more water than the capacity of all drains combined, the amount of water increases (And of course the mathematical drain acts as a negative tap so this example also works for a decrease).
Thank you for posting this, it was a great presentation and helped me review this material to help my nephew who is struggling with it just like I did 30 yrs ago. I now understand this (the math) better than ever! The physical concept has been clear long before tackling the formal math, but it's nice to be able to articulate the concept using the language of math.
Sorry... I will use my language to express my opinion about the video.... GRACIAS!!!!! Esto me llevó mucho esfuerzo de entender en mi época de estudiante de ingeniería. La claridad del video es perfecta. Muchas Gracias. Saludos desde La Ciudad del Humo.
So it's basically like saying, when you add up everything produced in a country and subtract everything consumed, the result is the exports of the country.
Great video ofcourse. But not the "deepest intuition", the deepest intuition of divergence can only be gained by properly understanding its derivation.
Maybe because we assume area implies 2D, while surface implies 2D area in 3D space. But yeah, the semantics are always weird. I've seen them used in so many different ways.
This is so simple, it reminds me of British comedian Alexei Sayle stating (incorrectly) Boyle's law for the expansion of gases. Alexei: "For a fixed mass of gas at fixed temperature and pressure, the volume remains constant. Alexei: "Well that's bloody obvious isn't it!"
Very well explained, good speaking rhythm, the visuals were great too, very easy to understand! Loved it I must confess I am revising this information years later I studied this in my college years, so it's easier to understand now. Good luck to my thermodynamics bros and braettes!
This is much easily understood while studying fluid dynamics... "The amount of water that flows through a closed surface, is equal to the change of volume inside the enclosed body"
Why is a single integral being used for the closed surface integral? Should it not be the closed double integral over the surface A of the dot product between F and the normal vector function to A times a surface element given by the length of the cross product times an area element.
It's not really a single integral, some people only write one integral, but it could be a double integral or a triple integral. What matters is over what surface you integrate. If you integrate over V, you know that it's gonna be a triple integral.
Thank you, Sami! #### Want more videos? #### As a channel member you have many cool benefits: ✅ Unlock ALL Physics Videos ✅ Channel Badges For Your Nickname ✅ Unique Channel Emojis ✅ Your Vote Counts 10x in Polls ✅ Immortalization in the Hall of Fame ✅ And that's not all! CLICK -> ru-vid.comjoin
I have a question. Didnt the tangential F components also came from the cube? But we omitted them through dot product. Why? Since F is a vector field, can it be perpendicular to cube at one side but be diagonal to another side? Your video made the consept a bit clearer so thank you.
is it like the flux passed by a volume of an object is equal to the flux passing through its continuous surface? i dont know much about this just watching for fun
